A cylindrical rod of length L insulated on its lateral

Chapter 6, Problem 6.73

(choose chapter or problem)

A cylindrical rod of length L insulated on its lateral surface is initially in contact at one end with a wall at temperature \(T_{H}\) and at the other end with a wall at a lower temperature \(T_{C}\). The temperature within the rod initially varies linearly with position z according to

\(T(z)=T_{\mathrm{H}}-\left(\frac{T_{\mathrm{H}}-T_{\mathrm{C}}}{L}\right) z\)

The rod is then insulated on its ends and eventually comes to a final equilibrium state where the temperature is \(T_{f}\). Evaluate \(T_{f}\) in terms of \(T_{H}\) and \(T_{C}\) and show that the amount of entropy produced is

\(\sigma=m c\left(1+\ln T_{\mathrm{f}}+\frac{T_{C}}{T_{\mathrm{H}}-T_{\mathrm{C}}} \ln T_{\mathrm{C}}-\frac{T_{\mathrm{H}}}{T_{\mathrm{H}}-T_{\mathrm{C}}} \ln T_{\mathrm{H}}\right)\)

where c is the specific heat of the rod.